Happy numbers are positive integers with the reciprocals of whose divisor sums t

Question

Happy numbers are positive integers with the reciprocals of whose divisor sums to 1.
Ex.: 4 (2*2....1/2+1/2=1), 27(3*3*3...1/3+1/3+1/3=1), and 32(2*4*4...1/2+1/4+1/4=1)
Extra happy numbers are positive integers that can be written as the sum of other positive integers whose reciprocals sum to 1.
Ex.: 4(2+2...1/2+1/2=1), 10(4+4+2...1/4+1/4+1/2=1), and 27(6+6+6+6+3....1/6+1/6+1/6+1/6+1/3=1)
A number that is neither happy or extra happy is unhappy.

1.) How many happy (but not extra happy) numbers are there?
I think that if a number is happy, it must be extra happy but don't know how to prove this..
2.) How many extra happy (but not happy) numbers are there?
I think this is infinite but I don't know how to prove this.
3.) How many unhappy numbers are there?
There are 13, they are 2,3,5,6,7,8,12,13,15,19,21,and23. Why does this stop at 23 tho? How do we prove this?
4.) How many numbers are both happy and extra happy?
I think this is infinite but I don't know how to prove this

Please explain this as thoroughly as possible.
Thanks! :)

Explanation / Answer

Hi I have done the solution for this in my notebook. But the solution is too big , I cannot type it here because very less time is remaining.. So please rate me Lifesaver and I'll share the answer with you through email or cramster inbox. I don't do this generally, but I have no other option here because there's very less time left... you need not worry as I have the solution ready

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